\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  28.0 & -4.00 x_{1} & -2.00 x_{2} &   & +  4.00 x_{4} & +  2.00 x_{5} & -1.00 x_{6} & -5.00 x_{7}\\
 x_{9}   &  24.0 & +  2.00 x_{1} & -2.00 x_{2} & -3.00 x_{3} & -2.00 x_{4} & -7.00 x_{5} & -7.00 x_{6} & -1.00 x_{7}\\
 x_{10}   &  -8.0 & +  2.00 x_{1} & +  4.00 x_{2} & +  1.00 x_{3} & -5.00 x_{4} & -3.00 x_{5} & -10.00 x_{6} & +  8.00 x_{7}\\
 x_{11}   &  47.0 & +  5.00 x_{1} & -7.00 x_{2} & -6.00 x_{3} & -5.00 x_{4} & +  3.00 x_{5} & +  9.00 x_{6} & -8.00 x_{7}\\
 x_{12}   &  0.0 & +  3.00 x_{1} & +  5.00 x_{2} & -8.00 x_{3} & -7.00 x_{4} & -10.00 x_{5} & +  6.00 x_{6} & +  3.00 x_{7}\\
 x_{13}   &  -1.0 & -5.00 x_{1} & +  2.00 x_{2} & -6.00 x_{3} & +  4.00 x_{4} & +  1.00 x_{5} &   & +  5.00 x_{7}\\
\hline
z    &  0.0 & -3.00 x_{1} & +  2.00 x_{2} & +  3.00 x_{3} &    &   & -1.00 x_{6} &   \\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  4.00 y_{8} & -2.00 y_{9} & -2.00 y_{10} & -5.00 y_{11} & -3.00 y_{12} & +  5.00 y_{13}\\
 y_{2}   &  1.0 & +  2.00 y_{8} & +  2.00 y_{9} & -4.00 y_{10} & +  7.00 y_{11} & -5.00 y_{12} & -2.00 y_{13}\\
 y_{3}   &  1.0  &   & +  3.00 y_{9} & -1.00 y_{10} & +  6.00 y_{11} & +  8.00 y_{12} & +  6.00 y_{13}\\
 y_{4}   &  1.0 & -4.00 y_{8} & +  2.00 y_{9} & +  5.00 y_{10} & +  5.00 y_{11} & +  7.00 y_{12} & -4.00 y_{13}\\
 y_{5}   &  1.0 & -2.00 y_{8} & +  7.00 y_{9} & +  3.00 y_{10} & -3.00 y_{11} & + 10.00 y_{12} & -1.00 y_{13}\\
 y_{6}   &  1.0 & +  1.00 y_{8} & +  7.00 y_{9} & + 10.00 y_{10} & -9.00 y_{11} & -6.00 y_{12} &   \\
 y_{7}   &  1.0 & +  5.00 y_{8} & +  1.00 y_{9} & -8.00 y_{10} & +  8.00 y_{11} & -3.00 y_{12} & -5.00 y_{13}\\
\hline
z    &  -0 & -28.00 y_{8} & -24.00 y_{9} & +  8.00 y_{10} & -47.00 y_{11} &   & +  1.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  0.75 & +  2.75 y_{8} & -2.25 y_{9} & +  0.25 y_{7} & -7.00 y_{11} & -2.25 y_{12} & +  6.25 y_{13}\\
 y_{2}   &  0.5 & -0.50 y_{8} & +  1.50 y_{9} & +  0.50 y_{7} & +  3.00 y_{11} & -3.50 y_{12} & +  0.50 y_{13}\\
 y_{3}   &  0.875 & -0.62 y_{8} & +  2.88 y_{9} & +  0.12 y_{7} & +  5.00 y_{11} & +  8.38 y_{12} & +  6.62 y_{13}\\
 y_{4}   &  1.625 & -0.88 y_{8} & +  2.62 y_{9} & -0.62 y_{7} & + 10.00 y_{11} & +  5.12 y_{12} & -7.12 y_{13}\\
 y_{5}   &  1.375 & -0.12 y_{8} & +  7.38 y_{9} & -0.38 y_{7} &   & +  8.88 y_{12} & -2.88 y_{13}\\
 y_{6}   &  2.25 & +  7.25 y_{8} & +  8.25 y_{9} & -1.25 y_{7} & +  1.00 y_{11} & -9.75 y_{12} & -6.25 y_{13}\\
 y_{10}   &  0.125 & +  0.62 y_{8} & +  0.12 y_{9} & -0.12 y_{7} & +  1.00 y_{11} & -0.38 y_{12} & -0.62 y_{13}\\
\hline
z    &  1.0 & -23.00 y_{8} & -23.00 y_{9} & -1.00 y_{7} & -39.00 y_{11} & -3.00 y_{12} & -4.00 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  23.0 & -2.75 x_{1} & +  0.50 x_{2} & +  0.62 x_{3} & +  0.88 x_{4} & +  0.12 x_{5} & -7.25 x_{6} & -0.62 x_{10}\\
 x_{9}   &  23.0 & +  2.25 x_{1} & -1.50 x_{2} & -2.88 x_{3} & -2.62 x_{4} & -7.38 x_{5} & -8.25 x_{6} & -0.12 x_{10}\\
 x_{7}   &  1.0 & -0.25 x_{1} & -0.50 x_{2} & -0.12 x_{3} & +  0.62 x_{4} & +  0.38 x_{5} & +  1.25 x_{6} & +  0.12 x_{10}\\
 x_{11}   &  39.0 & +  7.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & -10.00 x_{4} &   & -1.00 x_{6} & -1.00 x_{10}\\
 x_{12}   &  3.0 & +  2.25 x_{1} & +  3.50 x_{2} & -8.38 x_{3} & -5.12 x_{4} & -8.88 x_{5} & +  9.75 x_{6} & +  0.38 x_{10}\\
 x_{13}   &  4.0 & -6.25 x_{1} & -0.50 x_{2} & -6.62 x_{3} & +  7.12 x_{4} & +  2.88 x_{5} & +  6.25 x_{6} & +  0.62 x_{10}\\
\hline
z    &  -1.0 & -0.75 x_{1} & -0.50 x_{2} & -0.88 x_{3} & -1.62 x_{4} & -1.38 x_{5} & -2.25 x_{6} & -0.12 x_{10}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  23.0 & -2.75 x_{1} & +  0.50 x_{2} & +  0.62 x_{3} & +  0.88 x_{4} & +  0.12 x_{5} & -7.25 x_{6} & -0.62 x_{10}\\
 x_{9}   &  23.0 & +  2.25 x_{1} & -1.50 x_{2} & -2.88 x_{3} & -2.62 x_{4} & -7.38 x_{5} & -8.25 x_{6} & -0.12 x_{10}\\
 x_{7}   &  1.0 & -0.25 x_{1} & -0.50 x_{2} & -0.12 x_{3} & +  0.62 x_{4} & +  0.38 x_{5} & +  1.25 x_{6} & +  0.12 x_{10}\\
 x_{11}   &  39.0 & +  7.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & -10.00 x_{4} &   & -1.00 x_{6} & -1.00 x_{10}\\
 x_{12}   &  3.0 & +  2.25 x_{1} & +  3.50 x_{2} & -8.38 x_{3} & -5.12 x_{4} & -8.88 x_{5} & +  9.75 x_{6} & +  0.38 x_{10}\\
 x_{13}   &  4.0 & -6.25 x_{1} & -0.50 x_{2} & -6.62 x_{3} & +  7.12 x_{4} & +  2.88 x_{5} & +  6.25 x_{6} & +  0.62 x_{10}\\
\hline
z    &  0.0 & -3.00 x_{1} & +  2.00 x_{2} & +  3.00 x_{3} &    &   & -1.00 x_{6} &   \\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  23.0 & -2.75 x_{1} & +  0.50 x_{2} & +  0.62 x_{3} & +  0.88 x_{4} & +  0.12 x_{5} & -7.25 x_{6} & -0.62 x_{10}\\
 x_{9}   &  23.0 & +  2.25 x_{1} & -1.50 x_{2} & -2.88 x_{3} & -2.62 x_{4} & -7.38 x_{5} & -8.25 x_{6} & -0.12 x_{10}\\
 x_{7}   &  1.0 & -0.25 x_{1} & -0.50 x_{2} & -0.12 x_{3} & +  0.62 x_{4} & +  0.38 x_{5} & +  1.25 x_{6} & +  0.12 x_{10}\\
 x_{11}   &  39.0 & +  7.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & -10.00 x_{4} &   & -1.00 x_{6} & -1.00 x_{10}\\
 x_{12}   &  3.0 & +  2.25 x_{1} & +  3.50 x_{2} & -8.38 x_{3} & -5.12 x_{4} & -8.88 x_{5} & +  9.75 x_{6} & +  0.38 x_{10}\\
 x_{13}   &  4.0 & -6.25 x_{1} & -0.50 x_{2} & -6.62 x_{3} & +  7.12 x_{4} & +  2.88 x_{5} & +  6.25 x_{6} & +  0.62 x_{10}\\
\hline
z    &  0.0 & -3.00 x_{1} & +  2.00 x_{2} & +  3.00 x_{3} &    &   & -1.00 x_{6} &   \\
\end{array}\]


 $ x_{2} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  24.0 & -3.00 x_{1} & -1.00 x_{7} & +  0.50 x_{3} & +  1.50 x_{4} & +  0.50 x_{5} & -6.00 x_{6} & -0.50 x_{10}\\
 x_{9}   &  20.0 & +  3.00 x_{1} & +  3.00 x_{7} & -2.50 x_{3} & -4.50 x_{4} & -8.50 x_{5} & -12.00 x_{6} & -0.50 x_{10}\\
 x_{2}   &  2.0 & -0.50 x_{1} & -2.00 x_{7} & -0.25 x_{3} & +  1.25 x_{4} & +  0.75 x_{5} & +  2.50 x_{6} & +  0.25 x_{10}\\
 x_{11}   &  33.0 & +  8.50 x_{1} & +  6.00 x_{7} & -4.25 x_{3} & -13.75 x_{4} & -2.25 x_{5} & -8.50 x_{6} & -1.75 x_{10}\\
 x_{12}   &  10.0 & +  0.50 x_{1} & -7.00 x_{7} & -9.25 x_{3} & -0.75 x_{4} & -6.25 x_{5} & + 18.50 x_{6} & +  1.25 x_{10}\\
 x_{13}   &  3.0 & -6.00 x_{1} & +  1.00 x_{7} & -6.50 x_{3} & +  6.50 x_{4} & +  2.50 x_{5} & +  5.00 x_{6} & +  0.50 x_{10}\\
\hline
z    &  4.0 & -4.00 x_{1} & -4.00 x_{7} & +  2.50 x_{3} & +  2.50 x_{4} & +  1.50 x_{5} & +  4.00 x_{6} & +  0.50 x_{10}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  24.2307692308 & -3.46 x_{1} & -0.92 x_{7} & -0.08 x_{13} & +  2.00 x_{4} & +  0.69 x_{5} & -5.62 x_{6} & -0.46 x_{10}\\
 x_{9}   &  18.8461538462 & +  5.31 x_{1} & +  2.62 x_{7} & +  0.38 x_{13} & -7.00 x_{4} & -9.46 x_{5} & -13.92 x_{6} & -0.69 x_{10}\\
 x_{2}   &  1.88461538462 & -0.27 x_{1} & -2.04 x_{7} & +  0.04 x_{13} & +  1.00 x_{4} & +  0.65 x_{5} & +  2.31 x_{6} & +  0.23 x_{10}\\
 x_{11}   &  31.0384615385 & + 12.42 x_{1} & +  5.35 x_{7} & +  0.65 x_{13} & -18.00 x_{4} & -3.88 x_{5} & -11.77 x_{6} & -2.08 x_{10}\\
 x_{12}   &  5.73076923077 & +  9.04 x_{1} & -8.42 x_{7} & +  1.42 x_{13} & -10.00 x_{4} & -9.81 x_{5} & + 11.38 x_{6} & +  0.54 x_{10}\\
 x_{3}   &  0.461538461538 & -0.92 x_{1} & +  0.15 x_{7} & -0.15 x_{13} & +  1.00 x_{4} & +  0.38 x_{5} & +  0.77 x_{6} & +  0.08 x_{10}\\
\hline
z    &  5.15384615385 & -6.31 x_{1} & -3.62 x_{7} & -0.38 x_{13} & +  5.00 x_{4} & +  2.46 x_{5} & +  5.92 x_{6} & +  0.69 x_{10}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  25.3769230769 & -1.65 x_{1} & -2.61 x_{7} & +  0.21 x_{13} & -0.20 x_{12} & -1.27 x_{5} & -3.34 x_{6} & -0.35 x_{10}\\
 x_{9}   &  14.8346153846 & -1.02 x_{1} & +  8.51 x_{7} & -0.61 x_{13} & +  0.70 x_{12} & -2.60 x_{5} & -21.89 x_{6} & -1.07 x_{10}\\
 x_{2}   &  2.45769230769 & +  0.63 x_{1} & -2.88 x_{7} & +  0.18 x_{13} & -0.10 x_{12} & -0.33 x_{5} & +  3.45 x_{6} & +  0.28 x_{10}\\
 x_{11}   &  20.7230769231 & -3.85 x_{1} & + 20.51 x_{7} & -1.91 x_{13} & +  1.80 x_{12} & + 13.77 x_{5} & -32.26 x_{6} & -3.05 x_{10}\\
 x_{4}   &  0.573076923077 & +  0.90 x_{1} & -0.84 x_{7} & +  0.14 x_{13} & -0.10 x_{12} & -0.98 x_{5} & +  1.14 x_{6} & +  0.05 x_{10}\\
 x_{3}   &  1.03461538462 & -0.02 x_{1} & -0.69 x_{7} & -0.01 x_{13} & -0.10 x_{12} & -0.60 x_{5} & +  1.91 x_{6} & +  0.13 x_{10}\\
\hline
z    &  8.01923076923 & -1.79 x_{1} & -7.83 x_{7} & +  0.33 x_{13} & -0.50 x_{12} & -2.44 x_{5} & + 11.62 x_{6} & +  0.96 x_{10}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  23.2324749642 & -1.26 x_{1} & -4.73 x_{7} & +  0.41 x_{13} & -0.39 x_{12} & -2.69 x_{5} & +  0.10 x_{11} & -0.04 x_{10}\\
 x_{9}   &  0.77217453505 & +  1.59 x_{1} & -5.40 x_{7} & +  0.68 x_{13} & -0.52 x_{12} & -11.94 x_{5} & +  0.68 x_{11} & +  1.00 x_{10}\\
 x_{2}   &  4.67131616595 & +  0.22 x_{1} & -0.69 x_{7} & -0.02 x_{13} & +  0.09 x_{12} & +  1.14 x_{5} & -0.11 x_{11} & -0.04 x_{10}\\
 x_{6}   &  0.64234620887 & -0.12 x_{1} & +  0.64 x_{7} & -0.06 x_{13} & +  0.06 x_{12} & +  0.43 x_{5} & -0.03 x_{11} & -0.09 x_{10}\\
 x_{4}   &  1.30436337625 & +  0.77 x_{1} & -0.12 x_{7} & +  0.07 x_{13} & -0.04 x_{12} & -0.49 x_{5} & -0.04 x_{11} & -0.05 x_{10}\\
 x_{3}   &  2.26001430615 & -0.25 x_{1} & +  0.52 x_{7} & -0.12 x_{13} & +  0.01 x_{12} & +  0.22 x_{5} & -0.06 x_{11} & -0.05 x_{10}\\
\hline
z    &  15.4803290415 & -3.17 x_{1} & -0.44 x_{7} & -0.36 x_{13} & +  0.15 x_{12} & +  2.52 x_{5} & -0.36 x_{11} & -0.14 x_{10}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  23.0582420545 & -1.61 x_{1} & -3.51 x_{7} & +  0.25 x_{13} & -0.27 x_{12} & +  0.23 x_{9} & -0.05 x_{11} & -0.26 x_{10}\\
 x_{5}   &  0.0646723447594 & +  0.13 x_{1} & -0.45 x_{7} & +  0.06 x_{13} & -0.04 x_{12} & -0.08 x_{9} & +  0.06 x_{11} & +  0.08 x_{10}\\
 x_{2}   &  4.74529460515 & +  0.38 x_{1} & -1.21 x_{7} & +  0.04 x_{13} & +  0.04 x_{12} & -0.10 x_{9} & -0.04 x_{11} & +  0.05 x_{10}\\
 x_{6}   &  0.669948377949 & -0.06 x_{1} & +  0.44 x_{7} & -0.03 x_{13} & +  0.04 x_{12} & -0.04 x_{9} & -0.01 x_{11} & -0.06 x_{10}\\
 x_{4}   &  1.27235873831 & +  0.70 x_{1} & +  0.11 x_{7} & +  0.05 x_{13} & -0.01 x_{12} & +  0.04 x_{9} & -0.06 x_{11} & -0.10 x_{10}\\
 x_{3}   &  2.27411608471 & -0.22 x_{1} & +  0.43 x_{7} & -0.11 x_{13} & -0.00 x_{12} & -0.02 x_{9} & -0.05 x_{11} & -0.03 x_{10}\\
\hline
z    &  15.6429890865 & -2.84 x_{1} & -1.58 x_{7} & -0.22 x_{13} & +  0.04 x_{12} & -0.21 x_{9} & -0.22 x_{11} & +  0.08 x_{10}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  20.0503059143 & -1.33 x_{1} & -5.50 x_{7} & +  0.41 x_{13} & -0.44 x_{12} & +  0.39 x_{9} & -0.02 x_{11} & +  4.49 x_{6}\\
 x_{5}   &  1.01767505099 & +  0.04 x_{1} & +  0.18 x_{7} & +  0.01 x_{13} & +  0.01 x_{12} & -0.13 x_{9} & +  0.05 x_{11} & -1.42 x_{6}\\
 x_{2}   &  5.37049626105 & +  0.32 x_{1} & -0.80 x_{7} & +  0.01 x_{13} & +  0.08 x_{12} & -0.13 x_{9} & -0.05 x_{11} & -0.93 x_{6}\\
 x_{10}   &  11.4031271244 & -1.06 x_{1} & +  7.53 x_{7} & -0.59 x_{13} & +  0.63 x_{12} & -0.61 x_{9} & -0.11 x_{11} & -17.02 x_{6}\\
 x_{4}   &  0.188987083617 & +  0.80 x_{1} & -0.61 x_{7} & +  0.10 x_{13} & -0.07 x_{12} & +  0.10 x_{9} & -0.05 x_{11} & +  1.62 x_{6}\\
 x_{3}   &  1.91910265126 & -0.18 x_{1} & +  0.19 x_{7} & -0.09 x_{13} & -0.02 x_{12} & +  0.00 x_{9} & -0.04 x_{11} & +  0.53 x_{6}\\
\hline
z    &  16.4983004759 & -2.92 x_{1} & -1.02 x_{7} & -0.26 x_{13} & +  0.09 x_{12} & -0.26 x_{9} & -0.23 x_{11} & -1.28 x_{6}\\
\end{array}\]


 $ x_{12} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  18.9523809524 & -6.00 x_{1} & -1.95 x_{7} & -0.19 x_{13} & +  5.81 x_{4} & -0.19 x_{9} & +  0.29 x_{11} & -4.90 x_{6}\\
 x_{5}   &  1.04081632653 & +  0.14 x_{1} & +  0.10 x_{7} & +  0.02 x_{13} & -0.12 x_{4} & -0.12 x_{9} & +  0.04 x_{11} & -1.22 x_{6}\\
 x_{2}   &  5.56462585034 & +  1.14 x_{1} & -1.42 x_{7} & +  0.12 x_{13} & -1.03 x_{4} & -0.03 x_{9} & -0.10 x_{11} & +  0.73 x_{6}\\
 x_{10}   &  12.9977324263 & +  5.71 x_{1} & +  2.38 x_{7} & +  0.28 x_{13} & -8.44 x_{4} & +  0.23 x_{9} & -0.56 x_{11} & -3.38 x_{6}\\
 x_{12}   &  2.52154195011 & + 10.71 x_{1} & -8.14 x_{7} & +  1.37 x_{13} & -13.34 x_{4} & +  1.32 x_{9} & -0.70 x_{11} & + 21.58 x_{6}\\
 x_{3}   &  1.86167800454 & -0.43 x_{1} & +  0.38 x_{7} & -0.12 x_{13} & +  0.30 x_{4} & -0.03 x_{9} & -0.03 x_{11} & +  0.04 x_{6}\\
\hline
z    &  16.7142857143 & -2.00 x_{1} & -1.71 x_{7} & -0.14 x_{13} & -1.14 x_{4} & -0.14 x_{9} & -0.29 x_{11} & +  0.57 x_{6}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  14.7833333333 & -6.57 x_{1} & -2.36 x_{7} & -0.27 x_{13} & +  6.30 x_{4} & +  0.30 x_{9} & +  0.12 x_{11} & +  4.01 x_{5}\\
 x_{6}   &  0.85 & +  0.12 x_{1} & +  0.08 x_{7} & +  0.02 x_{13} & -0.10 x_{4} & -0.10 x_{9} & +  0.03 x_{11} & -0.82 x_{5}\\
 x_{2}   &  6.18333333333 & +  1.23 x_{1} & -1.36 x_{7} & +  0.13 x_{13} & -1.10 x_{4} & -0.10 x_{9} & -0.08 x_{11} & -0.59 x_{5}\\
 x_{10}   &  10.1277777778 & +  5.32 x_{1} & +  2.10 x_{7} & +  0.22 x_{13} & -8.10 x_{4} & +  0.57 x_{9} & -0.67 x_{11} & +  2.76 x_{5}\\
 x_{12}   &  20.8611111111 & + 13.23 x_{1} & -6.34 x_{7} & +  1.73 x_{13} & -15.50 x_{4} & -0.83 x_{9} & +  0.02 x_{11} & -17.62 x_{5}\\
 x_{3}   &  1.89444444444 & -0.42 x_{1} & +  0.38 x_{7} & -0.12 x_{13} & +  0.30 x_{4} & -0.03 x_{9} & -0.03 x_{11} & -0.03 x_{5}\\
\hline
z    &  17.2 & -1.93 x_{1} & -1.67 x_{7} & -0.13 x_{13} & -1.20 x_{4} & -0.20 x_{9} & -0.27 x_{11} & -0.47 x_{5}\\
\end{array}\]
Final Dictionary
Solution:  17.2
Num Pivots:  9
\end{document}
